A full year resident of NYC must paid taxes over all earned income, regardless where it was received, right? So, in case the person works out NYC or out NY state, is the employer suppose to withheld NYC tax and paid back to the NY city, which will avoid the taxpayer be in short at the moment of tax filing?

The answer is not that simple. It will depend upon where you worked and where you lived and in which state your employer is?

Thanks for answer.
Lives in NYC; works in Connecticut; and employer's state branch is NY.

If you live in NY city, then ALL earned income is subject to NY city tax during the period when you lived in NYC.